The disk complex and topologically minimal surfaces in the 3-sphere

TL;DR

This paper investigates the topological properties of Heegaard surfaces in the 3-sphere, revealing their disk complex structure and minimality index, which advances understanding of 3-manifold topology.

Contribution

It establishes that the disk complex of genus g > 1 Heegaard surfaces in the 3-sphere is homotopy equivalent to a wedge of spheres, determining their topological minimality index.

Findings

Disk complex is homotopy equivalent to a wedge of (2g-2)-dimensional spheres

Genus g > 1 Heegaard surfaces are topologically minimal with index 2g-1

Provides new insights into the topology of 3-sphere Heegaard splittings

Abstract

We show that the disk complex of a genus $g>1$ Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of $(2g-2)$-dimensional spheres. This implies that genus $g>1$ Heegaard surfaces for the 3-sphere are topologically minimal with index $2g-1$.